In this paper, by using the fixed points of strict-set-contractions, we study the existence of at least one or two positive solutions to the four-point boundary value problem y''(t)+a(t)f(y(t)=θ, 0<t<1, y(0)=μy(ξ), y(1)=βy(η) in Banach space E, where θ is zero element of E, 0<ξ≤η< 1, 0<μ< 1/1−ξ, 0<β<1/η and μξ(1-β)+(1-μ)(1-βη)>0. As an application, we also give one example to demonstrate our results.

With the help of the coincidence degree continuation theorem, the xistence of periodic solutions of a nonlinear second-order differential equation with deviating argument x''(t)+ƒ1(x(t)x'(t+ƒ2(x(t))(x'(t))2+g(x(t-r(t))=0, under some assumptions are obtained.

this paper, we give existence and uniqueness results for solutions of multipoint boundary value problems of the form x'=ƒ(t,x(t))+e(t), t∈(0,1), m∑j=1/Ajx (ηj)=0, where ƒ: [0, 1] x Rn→Rn is s Carathdodory function, Ajs (j=1, 2,……,m) are constant square matrices of order n1, 0<η1<η2<…<ηm-1<ηm<1, and e(t) ∈L1([0, 1], Rn). The existence of solutions is proven by the coincidence degree theory. As an application, we also give one example to demonstrate our results.

We consider the singular three-point boundary value problems (Φp (y'))'+a(t)ƒ(y (t))=0, 0<t<1, y' (0)=0, y (1)=βy (n), where Φp (s)=|s|p-2s, p>2, 0<β< 1, 0<η< 1, f ∈ C ((0,+∞), (0,+∞)), a: (0, 1)～(0,+∞), and has countably many singularities in (0, 1/2). We show that there exist countably many positive solutions by using the fixed-point index theory.

In this paper, we consider the following second order ordinary differential equation x''=ƒ(t, x (t),x'(t))+e(t), t∈(0, 1), (1.1) subject to one of the following boundary value conditions: x(0)=m-2∑i=1/αix(ξi), x(1)=n-2∑j=1/βjx(ηj), (1.2) x'(0)=m-2∑i=1/αix1 (ξi), x'(1)=n-2∑j=1/βjx1 (ηj), (1.3) x'(0)=m-2∑i=1/αix1 (ξi), x'(1)=n-2∑j=1/βjx (ηj), (1.4) where αi (1≤i≤m-2), βj (1≤j≤n-2) ∈ R, 0<ξ1<ξ2<……<ξm-2<1, 0<η1<η2<……<ηn-2< 1. When all the ais have no the same sign and all the bjs have no the same sign, some existence results are given for (1.1) with boundary conditions (1.2)-(1.4) at resonance case. We also give some examples to demonstrate our results.